Yıl 2021,
Cilt: 50 Sayı: 1, 199 - 215, 04.02.2021
İsmail Aydın
,
Ramazan Akgün
Kaynakça
- [1] G. Anatriello, Iterated grand and small Lebesgue spaces, Collect. Math. 65, 273–284,
2014.
- [2] R. Akgün, Trigonometric approximation of functions in generalized Lebesgue spaces
with variable exponent, Ukrainian Math. J. 63 (1), 1–26, 2011.
- [3] R. Akgün, Approximating polynomials for functions of weighted Smirnov-Orlicz
spaces, J. Funct. Spaces Appl. 2012, Art. ID 982360, 2012.
- [4] R. Akgün and D.M. Israfilov, Approximation and moduli of smoothness of fractional
order in Smirnov-Orlicz spaces, Glas. Mat. Ser. III, 42 (2), 121–136, 2008.
- [5] R. Akgün and D.M. Israfilov, Polynomial approximation in weighted Smirnov Orlicz
space, Proc. A. Razmadze Math. Inst. 139, 89–92, 2005.
- [6] R. Akgün and D.M. Israfilov, Approximation by interpolating polynomials in Smirnov-
Orlicz class, J. Korean Math. Soc. 43 (2), 413–424, 2006.
- [7] R. Akgün and D.M. Israfilov, Simultaneous and converse approximation theorems in
weighted Orlicz spaces, Bull. Belg. Math. Soc. 17 (2), 13–28, 2010.
- [8] R. Akgün and V. Kokilashvili, On converse theorems of trigonometric approximation
in weighted variable exponent Lebesgue spaces, Banach J. Math. Anal. 5 (1), 70-82,
2011.
- [9] I. Aydin, Weighted variable Sobolev spaces and capacity, J. Funct. Spaces Appl. 2012,
Art. ID 132690, 2012.
- [10] D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Har-
monic Analysis (Applied and Numerical Harmonic Analysis), Birkhäuser/Springer,
Heidelberg, 2013.
- [11] D. Cruz-Uribe, L. Diening and P. Hästö, The maximal operator on weighted variable
Lebesgue spaces, Fract. Calc. Appl. Anal. 14 (3), 361–374, 2011.
- [12] N. Danelia and V. Kokilashvili, Approximation by trigonometric polynomials in the
framework of variable exponent Lebesgue spaces, Georgian Math. J. 23 (1), 43–53,
2016.
- [13] L. Diening, Maximal function on generalized Lebesgue spaces Lp(.), Math. Inequal.
Appl. 7, 245–253, 2004.
- [14] L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev Spaces
with Variable Exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg,
2011.
- [15] V.K. Dzyadyk and I.A. Shevchuk, Theory of Uniform Approximation of Functions by
Polynomials, Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany, 2008.
- [16] A. Fiorenza, B. Gupta and P. Jain, The maximal theorem in weighted grand Lebesgue
spaces, Studia Math. 188 (2), 123–133, 2008.
- [17] A. Fiorenza, V. Kokilashvili and A. Meskhi, Hardy-Littlewood maximal operator in
weighted grand variable exponent Lebesgue space, Mediterr. J. Math. 14 (118), 2017.
- [18] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator,
Manuscripta Math. 92, 249–258, 1997.
- [19] D.M. Israfilov and R. Akgün, Approximation in weighted Smirnov-Orlicz classes, J.
Math. Kyoto Univ. 46 (4), 755–770, 2006.
- [20] D.M. Israfilov and A. Testici, Approximation in weighted generalized grand Lebesgue
spaces, Colloq. Math. 143, 113–126, 2016.
- [21] D.M. Israfilov and A. Testici, Approximation in weighted generalized grand Smirnov
spaces, Studia Sci. Math. Hungar. 54 (4), 471–488, 2017.
- [22] T. Iwaniec and C. Sbordone, On integrability of the Jacobien under minimal hypothe-
ses, Arch. Ration. Mech. Anal. 119, 129–143, 1992.
- [23] V. Kokilashvili and A. Meskhi, Maximal and Calderon-Zygmund operators in grand
variable exponent Lebesgue spaces, Georgian Math. J. 21, 447–461, 2014.
- [24] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41
(116), 592–618, 1991.
- [25] W. Orlicz. Über konjugierte exponentenfolgen, Stud. Math. 3, 200–211, 1931.
- [26] R.A. De Vore and G.G. Lorentz, Constructive Approximation, Springer, 1993.
- [27] A. Zygmund, Trigonometric Series, Volume I-II, Cambridge University Press, Cam-
bridge, 1968.
Weighted variable exponent grand Lebesgue spaces and inequalities of approximation
Yıl 2021,
Cilt: 50 Sayı: 1, 199 - 215, 04.02.2021
İsmail Aydın
,
Ramazan Akgün
Öz
In this paper we discuss and investigate trigonometric approximation in weighted grand variable exponent Lebesgue spaces. We also prove the direct and inverse theorems in these spaces.
Kaynakça
- [1] G. Anatriello, Iterated grand and small Lebesgue spaces, Collect. Math. 65, 273–284,
2014.
- [2] R. Akgün, Trigonometric approximation of functions in generalized Lebesgue spaces
with variable exponent, Ukrainian Math. J. 63 (1), 1–26, 2011.
- [3] R. Akgün, Approximating polynomials for functions of weighted Smirnov-Orlicz
spaces, J. Funct. Spaces Appl. 2012, Art. ID 982360, 2012.
- [4] R. Akgün and D.M. Israfilov, Approximation and moduli of smoothness of fractional
order in Smirnov-Orlicz spaces, Glas. Mat. Ser. III, 42 (2), 121–136, 2008.
- [5] R. Akgün and D.M. Israfilov, Polynomial approximation in weighted Smirnov Orlicz
space, Proc. A. Razmadze Math. Inst. 139, 89–92, 2005.
- [6] R. Akgün and D.M. Israfilov, Approximation by interpolating polynomials in Smirnov-
Orlicz class, J. Korean Math. Soc. 43 (2), 413–424, 2006.
- [7] R. Akgün and D.M. Israfilov, Simultaneous and converse approximation theorems in
weighted Orlicz spaces, Bull. Belg. Math. Soc. 17 (2), 13–28, 2010.
- [8] R. Akgün and V. Kokilashvili, On converse theorems of trigonometric approximation
in weighted variable exponent Lebesgue spaces, Banach J. Math. Anal. 5 (1), 70-82,
2011.
- [9] I. Aydin, Weighted variable Sobolev spaces and capacity, J. Funct. Spaces Appl. 2012,
Art. ID 132690, 2012.
- [10] D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Har-
monic Analysis (Applied and Numerical Harmonic Analysis), Birkhäuser/Springer,
Heidelberg, 2013.
- [11] D. Cruz-Uribe, L. Diening and P. Hästö, The maximal operator on weighted variable
Lebesgue spaces, Fract. Calc. Appl. Anal. 14 (3), 361–374, 2011.
- [12] N. Danelia and V. Kokilashvili, Approximation by trigonometric polynomials in the
framework of variable exponent Lebesgue spaces, Georgian Math. J. 23 (1), 43–53,
2016.
- [13] L. Diening, Maximal function on generalized Lebesgue spaces Lp(.), Math. Inequal.
Appl. 7, 245–253, 2004.
- [14] L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev Spaces
with Variable Exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg,
2011.
- [15] V.K. Dzyadyk and I.A. Shevchuk, Theory of Uniform Approximation of Functions by
Polynomials, Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany, 2008.
- [16] A. Fiorenza, B. Gupta and P. Jain, The maximal theorem in weighted grand Lebesgue
spaces, Studia Math. 188 (2), 123–133, 2008.
- [17] A. Fiorenza, V. Kokilashvili and A. Meskhi, Hardy-Littlewood maximal operator in
weighted grand variable exponent Lebesgue space, Mediterr. J. Math. 14 (118), 2017.
- [18] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator,
Manuscripta Math. 92, 249–258, 1997.
- [19] D.M. Israfilov and R. Akgün, Approximation in weighted Smirnov-Orlicz classes, J.
Math. Kyoto Univ. 46 (4), 755–770, 2006.
- [20] D.M. Israfilov and A. Testici, Approximation in weighted generalized grand Lebesgue
spaces, Colloq. Math. 143, 113–126, 2016.
- [21] D.M. Israfilov and A. Testici, Approximation in weighted generalized grand Smirnov
spaces, Studia Sci. Math. Hungar. 54 (4), 471–488, 2017.
- [22] T. Iwaniec and C. Sbordone, On integrability of the Jacobien under minimal hypothe-
ses, Arch. Ration. Mech. Anal. 119, 129–143, 1992.
- [23] V. Kokilashvili and A. Meskhi, Maximal and Calderon-Zygmund operators in grand
variable exponent Lebesgue spaces, Georgian Math. J. 21, 447–461, 2014.
- [24] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41
(116), 592–618, 1991.
- [25] W. Orlicz. Über konjugierte exponentenfolgen, Stud. Math. 3, 200–211, 1931.
- [26] R.A. De Vore and G.G. Lorentz, Constructive Approximation, Springer, 1993.
- [27] A. Zygmund, Trigonometric Series, Volume I-II, Cambridge University Press, Cam-
bridge, 1968.